Let A be the set of the elements of an algebra, and let E be an equivalence relation on the set A. The relation E is said to be compatible with an n-ary operation f, if for implies for any with. An equivalence relation compatible with all the operations of an algebra is called a congruence with respect to this algebra.
Quotient algebras and homomorphisms
Any equivalence relation E in a set A partitions this set in equivalence classes. The set of these equivalence classes is usually called the quotient set, and denoted A/E. For an algebra, it is straightforward to define the operations induced on the elements of A/E if E is a congruence. Specifically, for any operation of arity in define as, where denotes the equivalence class of generated by E. For an algebra, given a congruence E on, the algebra is called the quotient algebra of modulo E. There is a natural homomorphism from to mapping every element to its equivalence class. In fact, every homomorphismh determines a congruence relation via the kernel of the homomorphism,. Given an algebra, a homomorphism h thus defines two algebras homomorphic to, the image h and The two are isomorphic, a result known as the homomorphic image theorem or as the first isomorphism theorem for universal algebra. Formally, let be a surjective homomorphism. Then, there exists a unique isomorphism g from onto such that gcomposed with the natural homomorphism induced by equals h.
For every algebra on the set A, the identity relation on A, and are trivial congruences. An algebra with no other congruences is called simple. Let be the set of congruences on the algebra. Because congruences are closed under intersection, we can define a meet operation: by simply taking the intersection of the congruences. On the other hand, congruences are not closed under union. However, we can define the closure of any binary relationE, with respect to a fixed algebra, such that it is a congruence, in the following way:. Note that the closure of a binary relation depends on the operations in, not just on the carrier set. Now define as. For every algebra, with the two operations defined above forms a lattice, called the congruence lattice of.
Maltsev conditions
If two congruences permute with the composition of relations as operation, i.e., then their join is equal to their composition:. An algebra is called congruence-permutable if every pair of its congruences permutes; likewise a variety is said to be congruence-permutable if all its members are congruence-permutable algebras. In 1954, Anatoly Maltsev established the following characterization of congruence-permutable varieties: a variety is congruence permutableif and only ifthere exist a ternary term such that ; this is called a Maltsev term and varieties with this property are called Maltsev varieties. Maltsev's characterization explains a large number of similar results in groups, rings, quasigroups. Generically, such conditions are called Maltsev conditions. This line of research led to the Pixley–Wille algorithm for generating Maltsev conditions associated with congruence identities.